There are at least two statements which go by the name of Artin's conjecture.
If 
is any continuous complex
finite-dimensional group representation of
the absolute Galois group of a number
field, then Artin showed how to associate an Artin
L-function 
with it. These 
-series
directly generalize zeta functions and Dirichlet
L-series, and as a result of work by Richard Brauer, 
is known to extend to a meromorphic
function on the complex plane. Artin's first
conjecture states that it is in fact holomorphic,
i.e., has no poles, with the possible exception of a pole at 
(Artin 1924). Compare with the generalized
Riemann hypothesis, which deals with the locations of the zeros
of certain 
-series.
Artin's second conjecture states that every integer not equal to 
or a square number is a primitive
root modulo 
for infinitely many 
and proposes a natural density for the set
of such 
which is a rational multiple of a constant
known as Artin's constant. There is an analogous
theorem for functions instead of numbers which has been
proved by Billharz (Shanks 1993, p. 147).
-Reihen." Abh. Math. Sem. Univ. Hamburg 3,
89-108, 1924.Matthews, K. R. "A Generalization of Artin's
Conjecture for Primitive Roots." Acta Arith. 29, 113-146, 1976.Moree,
P. "A Note on Artin's Conjecture." Simon Stevin 67, 255-257,
1993.Ram Murty, M. "Artin's Conjecture for Primitive Roots."
Math. Intell. 10, 59-67, 1988.Shanks, D.